Critical points at infinity in Yamabe changing-sign equations

In the well-known paper [A. Bahri and J.M. Coron, Commun. Pure Appl. Math. 41 (1988) 253–294], Bahri and Coron develop the theory of critical points at innity and find the solutions of Yamabe problem via Morse theory. This is a very delicate problem because of the lack of compactness caused by the invariance under the conformal group. To obtain the desired results, one needs a careful analysis on the change of the topology of the level sets. In this work, the author continues to use these ideas and give a preliminary study of the topological features for the Yamabe sign-changing variational problem on domains of R3 or on spheres S3. One of key points consists to understand the Morse relations at innity based on the expansion of the energy functional in a neighborhood of innity. In particular, one study weather the relation ∂∞°∂∞ = 0 holds where ∂∞ is the intersection operator at innity. Although I could not understand completely the details, I believe such study is very delicate and the ideas and techniques developed could be also useful in the others context, in particular, some conformal invariant problems like Yang-Mills equations and harmonic maps. I recommend strongly the publication of the paper